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This paper is concerned with non-trivial solvability in $p$ -adic integers of systems of additive forms. Assuming that the congruence equation $a x^{k} + b y^{k} + c z^{k} \equiv d (m o d p)$ has a solution with $x y z ≢ 0 (m o d p)$ we have proved that any system of $R$ additive forms of degree $k$ with at least $2 \cdot 3^{R - 1} \cdot k + 1$ variables, has always non-trivial $p$ -adic solutions, provided $p ∤ k$ . The assumption of the solubility of the above congruence equation is guaranteed, for example, if $p > k^{4}$ .

On simultaneous diagonal equations and inequalities

J. Brüdern, R. J. Cook (1992)

Acta Arithmetica

On the congruence $a_{1} {(x_{1})}^{k} + . . . + a_{s} {(x_{s})}^{k} = N (m o d p^{n})$

J. Bovey (1973)

Acta Arithmetica

On the quasi-periodic $p$ -adic Ruban continued fractions

Basma Ammous, Nour Ben Mahmoud, Mohamed Hbaib (2022)

Czechoslovak Mathematical Journal

We study a family of quasi periodic $p$ -adic Ruban continued fractions in the $p$ -adic field $ℚ_{p}$ and we give a criterion of a quadratic or transcendental $p$ -adic number which based on the $p$ -adic version of the subspace theorem due to Schlickewei.

On zeros of diagonal forms over p-adic fields

Yismaw Alemu (1987)

Acta Arithmetica

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